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attractor (Definition)

Let

$\displaystyle \dot{x}=f(x)$
be a system of autonomous ordinary differential equation in $ \mathbb{R}^n$ defined by a vector field $ f\colon \mathbb{R}^n\to \mathbb{R}^n$. A set $ A$ is said to be an attracting set[GH,P] if
  1. $ A$ is closed and invariant,
  2. there exists an open neighborhood $ U$ of $ A$ such that all solution with initial solution in $ U$ will eventually enter $ A$ ($ x(t)\to A$) as $ t\to \infty$.
Additionally, if $ A$ contains a dense orbit then $ A$ is said to be an attractor[GH,P].
Conversely, a set $ R$ is said to be a repelling set[GH] if $ R$ satisfy the condition 1. and 2. where $ t\to \infty$ is replaced by $ t\to -\infty$. Similarly, if $ R$ contains a dense orbit then $ R$ is said to be a repellor[GH].

References

GH
GUCKENHEIMER, JOHN & HOLMES, PHILIP, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, New York, 1983.
P
PERKO, LAWRENCE, Differential Equations and Dynamical Systems, Springer, New York, 2001.



"attractor" is owned by Daume.
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Also defines:  attracting set, repelling set, repellor
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Cross-references: orbit, dense, contains, eventually, solution, neighborhood, open, invariant, closed, vector field, ordinary differential equation, autonomous
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This is version 1 of attractor, born on 2005-05-20.
Object id is 7089, canonical name is Attractor.
Accessed 3747 times total.

Classification:
AMS MSC34C99 (Ordinary differential equations :: Qualitative theory :: Miscellaneous)

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